x < x < 24 100 x < x - > 12 30. x - > - ExamExxpert

Class XI – NCERT – Maths Chapter 6

Linear Inequalities

6. Linear Inequalities

Exercise 6.1

Question 1:

Solve 24 100x , when (i) x is a natural number (ii) x is an integer.

Solution 1:

The given inequality is 24 100x .

24 100x

24 100

24 24

x [Dividing both sides by same positive number]

25

6x

(i) It is evident that 1, 2, 3 and 4 are the only natural numbers less than 25

6

Thus, when x is a natural number, the solutions of the given inequality are 1, 2, 3 and 4

Hence, in this case, the solution set is 1,2,3,4 .

(ii) The integers less than 25

6 are …. 3, 2, 1,0,1,2,3,4 .

Thus, when x is an integer , the solutions of the given inequality are … 3, 2, 1,0,1,2,3,4

Hence, in this case, the solution set is 3, 2, 1,0,1,2,3,4

Question 2:

Solve 12 30,x when

(i) x is a natural number

(ii) x is an integer

Solution 2:

The given inequality is 12 30.x

12 30.x

12 30

12 12

x

[Dividing both sides by same negative number]

5

2x

(i) There is no natural number less than 5

2

.

Thus, when x is a natural number, there is no solution of the given inequality.

(ii) The integers less than 5

2

are … 5, 4, 3 .

Thus, when x is an integer, the solutions of the given inequality are …., 5, 4, 3

Hence, in this case, the solution set is ...., 5, 4, 3 .


Linear Inequalities


Question 3:

Solve 5 3 7x , when

(i) x is an integer

(ii) x is a real number

Solution 3:

The given inequality is 5 3 7x .

5 3 7x

5 3 3 7 3x

5 10x

5 10

5 5

x

2x

(i) The integers less than 2 are …, 4, 3, 2, 1,0,1 .

Thus, when x is an integer, the solutions of the given inequality are …, 4, 3, 2, 1,0,1 .

Hence, in this case, the solution set is ..., 4, 3, 2, 1,0,1 .

(ii) When x is a real number, the solutions of the given inequality are given by 2,x that is, all

real numbers x which are less than 2.

Thus, the solution set of the given inequality is ,2x .

Question 4:

Solve 3 8 2x , when

(i) x is an integer

(ii) x is a real number

Solution 4:

The given inequality is 3 8 2x

3 8 2x

3 8 8 2 8x

3 6x

3 6

3 3

x

2x

(i) The integers greater than 2 are 1,0,1,2,...

Thus, when x is an integer, the solutions of the given inequality are 1,0,1,2,...

Hence, in this case, the solution set is 1,0,1,2,... .

(ii) When x is a real number, the solutions of the given inequality are all the real numbers, which

are greater than 2 .

Thus, in this case, the solution set is 2, .

https://www.ncertbooks.guru/cbse-ncert-solutions-pdf/


Linear Inequalities


Question 5:

Solve the given inequality for real : 4 3 5 7x x x

Solution 5:

4 3 5 7x x

4 3 7 5 7 7x x

4 4 5x x

4 4 4 5 4x x x x

4 x

Thus, all real numbers x, which are greater than 4 , are the solutions of the given inequality.

Hence, the solution set of the given inequality is 4, .

Question 6:

Solve the given inequality for real : 3 7 5 1x x x

Solution 6:

3 7 5 1x x

3 7 7 5 1 7x x

3 5 6x x

3 5 5 6 5x x x x

2 6x

2 6

2 2

x

3x

Thus, all real numbers x, which are less than 3 , are the solutions of the given inequality.

Hence, the solution set of the given inequality is , 3 .

Question 7:

Solve the given inequality for real :3 1 2 3x x x

Solution 7:

3 1 2 3x x

3 3 2 6x x

3 3 3 2 6 3x x

3 2 3x x

3 2 2 3 2x x x x

3x

Thus, all real numbers x, which are less than or equal to 3 , are the solutions of the given

inequality.



Linear Inequalities


Question 8:

Solve the given inequality for real :3 2 2 1x x x

Solution 8:

3 2 2 1x x

6 3 2 2x x

6 3 2 2 2 2x x x x

6 2x

6 6 2 6x

4x

4x

Thus, all real numbers x, which are greater than or equal to 4, are the solutions of the given

inequality.

Hence, the solution set of the given inequality is ( ,4] .

Question 9:

Solve the given inequality for real : 112 3

x xx x

Solution 9:

112 3

x xx

1 11 11

2 3x

6 3 211

6x

1111

6

x

11 11

6 11 11

x

16

x

6x

Thus, all real numbers x, which are less than 6, are the solutions of the given inequality.

Hence, the solution set of the given inequality is ,6 .

Question 10:

Solve the given inequality for real : 13 2

x xx

Solution 10:


Linear Inequalities


13 2

x x

13 2

x x

2 31

6

x x

16

x

6x

6x

Thus, all real numbers x, which are less than 6 , are the solutions of the given inequality.


Question 11:

Solve the given inequality for real 3 2 5 2

:5 3

x xx

Solution 11:

3 2 5 2

5 3

x x

9 2 25 2x x

9 18 50 25x x

9 18 25 50x x

34 18 50x

34 50 18x

34 68x

34 68

34 34

x

2x

Thus, all real numbers x, which are less than or equal to 2, are the solutions of the given hence

the solution set of the given inequality is ( ,2] .

Question 12:

Solve the given inequality for real 1 3 1

: 4 62 5 3

xx x

Solution 12:

1 3 1

4 62 5 3

xx


Linear Inequalities


3

3 4 2 65

xx

912 2 12

5

xx

912 12 2

5

xx

10 924

5

x x

245

x

120 x

Thus, all real numbers x, which are less than or equal to 120, are the solutions of the given

inequality.


Question 13:

Solve the given inequality for real : 2 2 3 10 6 2x x x

Solution 13:

2 2 3 10 6 2x x

4 6 10 6 12x x

4 4 6 12x x

4 12 6 4x x

8 2x

4 x

Thus, all real numbers x, which are greater than 4, are the solutions of the given inequality.

Hence, the solution set of the given inequality is (4, ) .

Question 14:

Solve the given inequality for real : 37 3 5 9 8 3x x x x

Solution 14:

37 3 5 9 8 3x x x

37 3 5 9 8 24x x x

32 3 24x x

32 24 3x x

8 4x

2 x


inequality.



Linear Inequalities


Question 15:

Solve the given inequality for real 5 2 7 3

:4 3 5

x xxx

Solution 15:

5 2 7 3

4 3 5

x xx

5 5 2 3 7 3

4 15

x xx

25 10 21 9

4 15

x x x

4 1

4 15

x x

15 4 4 1x x

15 16 4x x

4 16 15x x

4 x

Thus, all real numbers x, which are greater than 4, are the solutions of the given inequality.

Hence, the solution set of the given inequality is 4, .

Question 16:

Solve the given inequality for real 2 1 3 2 2

:3 4 5

x x xx

Solution 16:

2 1 3 2 2

3 4 5

x x x

2 1 5 3 2 4 2

3 20

x x x

2 1 15 10 8 4

3 20

x x x

2 1 19 18

3 20

x x

20 2 1 3 19 18x x

40 20 57 54x x

20 54 57 40x x

34 17x

2 x


Linear Inequalities



inequality.


Question 17:

Solve the given inequality and show the graph of the solution on number line:

3 2 2 1x x

Solution 17:

3 2 2 1x x

3 2 1 2x x

3x

The graphical representation of the solutions of the given inequality is as follows:

Question 18:


5 3 3 5x x

Solution 18:

5 3 3 5x x

5 3 5 3x x

2 2x

2 2

2 2

x

1x

The graphical representation of the solutions of the given inequality is as follows.

Question 19:


3 1 2 4x x

Solution 19:

3 1 2 4x x

3 3 2 8x x

3 8 2 3x x


Linear Inequalities


5 5x

5 5

5 5

x

1 x

The graphical representation of the solutions of the given inequality is as follows:

Question 20:


5 2 7 3

2 3 5

x xx

Solution 20:

5 2 7 3

2 3 5

x xx

5 5 2 3 7 3

2 15

x xx

25 10 21 9

2 15

x x x

4 1

2 15

x x

15 2 4 1x x

15 8 2x x

15 8 8 2 8x x x x

7 2x

2

7x

The graphical representation of the solutions of the given inequality is as follows.

Question 21:

Ravi obtained 70 and 75 marks in first two-unit test. Find the minimum marks he should get in

the third test to have an average of at least 60 marks.

Solution 21:

Let x be the marks obtained by Ravi in the third unit test.

Since the student should have an average of at least 60 marks,

70 7560

3

x


Linear Inequalities


145 180x

180 145x

35x

Thus, the student must obtain a minimum of 35 marks to have an average of at least 60 marks.

Question 22:

To receive Grade ‘A’ in a course, one must obtain an average of 90 marks or more in five

examinations (each of 100 marks). If Sunita’s marks in first four examinations are 87, 92, 94

and 95, find minimum marks that Sunita must obtain in fifth examination to get grade ‘A’ in the

course.

Solution 22:

Let x be the marks obtained by Sunita in the fifth examination.

In order to receive grade ‘A’ in the course, she must obtain an average of 90 marks or more in

five examinations.

Therefore,

87 92 94 9590

5

x

36890

5

x

368 450x

450 368x

82x

Thus, Sunita must obtain greater than or equal to 82 marks in the fifth examination.

Question 23:

Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that

their sum is more than 11.

Solution 23:

Let x be the smaller of the two consecutive odd positive integers. Then, the other integer is 2.x

Since both the integers are smaller than 10,

2 10x

10 2x

8 .....x i

Also, the sum of the two integers is more than 11.

2 11x x

2 2 11x

2 11 2x

2 9x

9

2x

4.5 .......x ii


Linear Inequalities


From (i) and (ii), we obtain

Since x is an odd number, x can take the values, 5 and 7.

Thus, the required possible pairs are (5, 7) and (7, 9).

Question 24:

Find all pairs of consecutive even positive integers, both of which are larger than 5 such that

their sum is less than 23.

Solution 24:

Let x be the smaller of the two consecutive even positive integers. Then, the other integer is

2x .

Since both the integers are larger than 5,

5........ 1x

Also, the sum of the two integers is less than 23

2 23x x

2 2 23x

2 23 2x

2 21x

21

2x

10.5 ...... 2x

From (1) and (2), we obtain 5 10.5x .

Since x is an even number, x can take the values, 6, 8 and 10.

Thus, the required possible pairs are 6,8 , 8,10 and 10,12 .

Question 25:

The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than

the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the

shortest side.

Solution 25:

Let the length of the shortest side of the triangle be x cm.

Then, length of the longest side = 3x cm

Length of the third side 3 2x cm

Since the perimeter of the triangle is at least 61 cm,

3 3 2 61xcm xcm x cm cm

7 2 61x

7 61 2x

7 63x


Linear Inequalities


7 63

7 7

x

9x

Thus, the minimum length of the shortest side is 9 cm.

Question 26:

A man wants to cut three lengths from a single piece of board of length 91 cm. The second length

is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest.

What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer

than the second?

[Hint: It x is the length of the shortest board, then , 3x x and 2x are the lengths of the second

and third piece, respectively. Thus, 3 2 91x x x and 2 3 5x x ]

Solution 26:

Let the length of the shortest piece be x cm. Then, length of the second piece and the third piece

are (x + 3) cm and 2x cm respectively.

Since the three lengths are to be cut from a single piece of board of length 91 cm,

3 2 91xcm x cm x cm cm

4 3 91x

4 91 3x

4 88x

4 88

4 4

x

22..... 1x

Also, the third piece is at least 5 cm longer than the second piece.

2 3 5x x

2 8x x

8 .... 2x

From (1) and (2), we obtain

8 22x

Thus, the possible length of the shortest board is greater than or equal to 8 cm but less than or

equal to 22 cm.

Exercise 6.2

Question 1:

Solve the given inequality graphically in two-dimensional plane: 5x y

Solution 1:


Linear Inequalities


The graphical representation of 5x y is given as dotted line in the figure below.

This line divides the xyplane in two half planes, I and II.

Select a point (not on the line), which lies in one of the half planes, to determine whether the

point satisfies the given inequality or not.

We select the point as (0, 0).

It is observed that,

0 0 5 or 0 5 , which is true

Therefore, half plane II is not the solution region of the given inequality. Also, it is evident that

any point on the line does not satisfy the given strict inequality.

Thus, the solution region of the given inequality is the shaded half plane I excluding the points

on the line.

This can be represented as follows.

Question 2:

Solve the given inequality graphically in two-dimensional plane: 2 6x y

Solution 2:

The graphical representation of 2 6x y is given in the figure below.

This line divides the xy-plane in two half planes, I and II.





2 0 0 6 or 0 6, which is false

Therefore, half plane I is not the solution region of the given inequality. Also, it is evident that

any point on the line satisfies the given inequality.

Thus, the solution region of the given inequality is the shaded half plane II including the points

on the line.



Linear Inequalities


Question 3:

Solve the given inequality graphically in two-dimensional plane: 3 4 12x y

Solution 3:

3 4 12x y

The graphical representation of 3 4 12x y is given in the figure below.

This line divides the xy-plane in two half planes, I and II.




Thus, the solution region of the given inequality is the shaded half plane I including the points

on the line.


Question 4:

Solve the given inequality graphically in two-dimensional plane: 8 2y x

Solution 4:

The graphical representation of 8 2y x is given in the figure below.


Linear Inequalities


This line divides the xy-plane in two half planes.





0 8 2 0 or 8 0 , which is true

Therefore, lower half plane is not the solution region of the given inequality. Also, it is evident

that any point on the line satisfies the given inequality.

Thus, the solution region of the given inequality is the half plane containing the point (0, 0)

including the line.

The solution region is represented by the shaded region as follows.

Question 5:

Solve the given inequality graphically in two-dimensional plane: 2x y

Solution 5:

The graphical representation of 2x y is given in the figure below.






0 0 2 or 0 2, which is true

Therefore, the lower half plane is not the solution region of the given inequality. Also, it is clear

that any point on the line satisfies the given inequality.


including the line.



Linear Inequalities


Question 6:


Solution 6:

The graphical representation of 2 3 6x y is given as dotted line in the figure below.






2 0 3 0 6 or 0 6, which is false

Therefore, the upper half plane is not the solution region of the given inequality. Also, it is clear

that any point on the line does not satisfy the given inequality.

Thus, the solution region of the given inequality is the half plane that does not contain the point

(0, 0) including the line.



Linear Inequalities


Question 7:


Solution 7:

The graphical representation of 3 2 6x y is given in the figure below.






3 0 2 0 6 or 0 6, which is true

Therefore, the lower half plane is not the solution region of the given inequality. Also, it is

evident that any point on the line satisfies the given inequality.


including the line.


Question 8:

Solve the given inequality graphically in two-dimensional plane: 3 5 30y x

Solution 8:

The graphical representation of 3 5 30y x is given as dotted line in the figure below.






3 0 5 0 30 or 0 30, which is true

Therefore, the upper half plane is not the solution region of the given inequality. Also, it is

evident that any point on the line does not satisfy the given inequality.


Linear Inequalities



excluding the line.


Question 9:

Solve the given inequality graphically in two-dimensional plane: 2y

Solution 9:

The graphical representation of 2y is given as dotted line in the figure below. This line

divides the xy-plane in two half planes.





0 2, which is false

Also, it is evident that any point on the line does not satisfy the given inequality.

Hence, every point below the line, 2y (excluding all the points on the line), determines the

solution of the given inequality.



Linear Inequalities


Question 10:

Solve the given inequality graphically in two-dimensional plane: 3x

Solution 10:

The graphical representation of 3x is given as dotted line in the figure below. This line

divides the xy-plane in two half planes.





0 3, which is true

Also, it is evident that any point on the line does not satisfy the given inequality.

Hence, every point on the right side of the line, 3x (excluding all the points on the line),

determines the solution of the given inequality.


Exercise 6.3

Question 1:

Solve the following system of inequalities graphically: 3, 2x y .

Solution 1:

3....... 1x

2....... 2y

The graph of the lines, 3x and 2,y are drawn in the figure below.

Inequality (1) represents the region on the right hand side of the line, 3x (including the line

3x ), and inequality (2) represents the region above the line, 2y (including the line y = 2).

Hence, the solution of the given system of linear inequalities is represented by the common

shaded region including the points on the respective lines as follows.


Linear Inequalities


Question 2:

Solve the following system of inequalities graphically: 3 2 12, 1, 2x y x y

Solution 2:

3 2 12........ 1x y

1....... 2x

2....... 3y

The graphs of the lines, 3 2 12, 1x y x , and 2y , are drawn in the figure below.

Inequality (1) represents the region below the line, 3 2 12x y (including the line

3 2 12).x y Inequality (2) represents the region on the right side of the line, 1x (including

the line 1x ). Inequality (3) represents the region above the line, 2y (including the line

2y ).




Linear Inequalities


Question 3:

Solve the following system of inequalities graphically: 2 6, 3 4 12x y x y

Solution 3:

2 6....... 1x y

3 4 12....... 2x y

The graph of the lines, 2 6x y and 3 4 12x y , are drawn in the figure below. Inequality

(1) represents the region above the line, 2 6x y (including the line 2 6x y ), and inequality

(2) represents the region below the line, 3 4 12x y (including the line 3 4 12x y ).





Linear Inequalities


Question 4:

Solve the following system of inequalities graphically: 4, 2 0x y x y

Solution 4:

4....... 1

2 0........ 2

x y

x y

The graph of the lines, 4x y and 2 0x y , are drawn in the figure below.

Inequality (1) represents the region above the line, 4x y (including the line 4x y ). It is

observed that (1, 0) satisfies the inequality, 2 0x y . 2 1 0 2 0

Therefore, inequality (2) represents the half plane corresponding to the line, 2 0x y ,

containing the point (1, 0) [excluding the line 2 0x y ].


shaded region including the points on line 4x y and excluding the points on line 2 0x y

as follows.


Linear Inequalities


Question 5:

Solve the following system of inequalities graphically: 2 1, 2 1x y x y

Solution 5:

2 1........ 1x y

2 1....... 2x y

The graph of the lines, 2 1x y and 2 1x y , are drawn in the figure below.

Inequality (1) represents the region below the line, 2 1x y (excluding the line 2 1x y ), and

inequality (2) represents the region above the line, 2 1x y (excluding the line 2 1).x y


shaded region excluding the points on the respective lines as follows.

Question 6:

Solve the following system of inequalities graphically: 6, 4x y x y

Solution 6:


Linear Inequalities


6 ....... 1x y

4...... 2x y

The graph of the lines, 6x y and 4x y , are drawn in the figure below.

Inequality (1) represents the region below the line, 6x y (including the line 6x y ), and

inequality (2) represents the region above the line, 4x y (including the line 4x y ).



Question 7:

Solve the following system of inequalities graphically: 2 8, 2 10x y x y

Solution 7:

2 8....... 1x y

2 10........ 2x y

The graph of the lines, 2 8x y and 2 10x y , are drawn in the figure below.

Inequality (1) represents the region above the line, 2 8x y , and inequality (2) represents the

region above the line, 2 10x y .




Linear Inequalities


Question 8:

Solve the following system of the inequalities graphically: 9, , 0x y y x x

Solution 8:

9....... 1x y

........ 2y x

0 ........ 3x

The graph of the lines, 9x y and y x , are drawn in the figure below.

Inequality (1) represents the region below the line, 9x y (including the line 9x y ). It is

observed that (0, 1) satisfies the inequality, . 1 0y x . Therefore, inequality (2) represents

the half plane corresponding to the line, y x , containing the point (0, 1) [excluding the line

y x ]. Inequality (3) represents the region on the right hand side of the line, 0x or y axis

(including axisy )


shaded region including the points on the lines, 9x y and 0x , and excluding the points

on line y x as follows.


Linear Inequalities


Question 9:

Solve the following system of inequalities graphically: 5 4 20, 1, 2x y x y

Solution 9:

5 4 20 ....... 1x y

1........ 2x

2....... 3y

The graph of the lines, 5 4 20, 1x y x and 2y , are drawn in the figure below.


5 4 20).x y Inequality (2) represents the region on the right hand side of the line, 1x

(including the line 1x ). Inequality (3) represents the region above the line, 2y (including

the line 2y ).




Linear Inequalities


Question 10:

Solve the following system of inequalities graphically: 3 4 60, 3 30, 0, 0x y x y x y

Solution 10:

3 4 60 ....... 1x y

3 30....... 2x y

The graph of the lines, 3 4 60x y and 3 30x y , are drawn in the figure below.


3 4 60),x y and inequality (2) represents the region below the line, 3 30x y (including the

line 3 30x y ).

Since 0x and 0y , every point in the common shaded region in the first quadrant including

the points on the respective line and the axes represents the solution of the given system of linear

inequalities.


Linear Inequalities


Question 11:

Solve the following system of inequalities graphically: 2 4, 3, 2 3 6x y x y x y

Solution 11:

2 4....... 1x y

3....... 2x y

2 3 6....... 3x y

The graph of the lines, 2 4, 3x y x y and 2 3 6x y , are drawn in the figure below.

Inequality (1) represents the region above the line, 2 4x y (including the line 2 4x y ).

Inequality (2) represents the region below the line, 3x y (including the line 3x y ).

Inequality (3) represents the region above the line, 2 3 6x y (including the line 2 3 6x y ).




Linear Inequalities


Question 12:

Solve the following system of inequalities graphically:

2 3, 3 4 12, 0, 1x y x y x y

Solution 12:

2 3....... 1x y

3 4 12........ 2x y

1..... 3y

The graph of the lines, 2 3, 3 4 12x x y and 1y , are drawn in the figure below.

Inequality (1) represents the region above the line, 2 3x y (including the line 2 3x y ).

Inequality (2) represents region above the line, 3 4 12x y (including the line 3 4 12x y ).

Inequality (3) represents the region above the line, 1y (including the line 1y ). The

inequality, 0x , represents the region on the right and side of y axis (including y axis).


shaded region including the points on the respective lines and y axis as follows.


Linear Inequalities


Question 13:


4 3 60, 2 , 3, , 0x y y x x x y

Solution 13:

4 3 60...... 1x y

2 ...... 2y x

3....... 3x

The graph of the lines, 4 3 60, 2x y y x 3x , are drawn in the figure below.


4 3 60).x y Inequality (2) represents the region above the line, 2y x (including the line

2y x ). Inequality (3) represents the region on the right hand side of the line, 3x (including

the line 3x ).




Linear Inequalities


Question 14:


3 2 150, 4 80, 15, 0, 0x y x y x y x

Solution 14:

3 2 150 ..... 1x y

4 80...... 2x y

15..... 3x

The graph of the lines, 3 2 150, 4 80x y x y and 15x , are drawn in the figure below.


3 2 150x y ). Inequality (2) represents the region below the line, 4 80x y (including the

line 4 80x y ). Inequality (3) represents the region on the left hand side the line, 15x

(including the line 15x ).


the points on the respective lines and the axes represents the solution of the given system of

linear inequalities.


Linear Inequalities


Question 15:


2 10, 1, 0, 0, 0x y x y x y x y

Solution 15:

2 10 ....... 1x y

1...... 2x y

0..... 3x y

The graph of the lines, 2 10, 1x y x y and 0x y , are drawn in the figure below.

Inequality (1) represents the region below the line, 2 10x y (including the line 2 10x y ).

Inequality (2) represents the region above the line, 1x y (including the line 1x y ).

Inequality (3) represents the region above the line, 0x y (including the line 0x y ).


the points on the respective lines and the axes represents the solution of the given system of

linear inequalities.


Linear Inequalities


Miscellaneous Exercise

Question 1:

Solve the inequality 2 3 4 5x

Solution 1:

2 3 4 5x

2 4 3 4 4 5 4x

6 3 9x

2 3x

Thus, all the real numbers, x, which are greater than or equal to 2 but less than or equal to 3, are

the solutions of the given inequality. The solution set for the given inequality is [2, 3].

Question 2:

Solve the inequality 6 3 2 4 12x

Solution 2:

6 3 2 4 12x

2 2 4 4x

2 2 4 4x

4 2 2 4 4x

2 2 0x

1 0x


Linear Inequalities


Thus, the solution set for the given inequality is 0,1 .

Question 3:

Solve the inequality 7

3 4 182

x

Solution 3:

73 4 18

2

x

73 4 18 4

2

x

77 14

2

x

77 14

2

x

1 22

x

2 4x


Question 4:

Solve the inequality 3 2

15 05

x

Solution 4:

3 215 0

5

x

75 3 2 0x

25 2 0x

25 2 2x

23 2x


Question 5:

Solve the inequality 3

12 4 25

x

Solution 5:

312 4 2

5

x


Linear Inequalities


312 4 2 4

5

x

316 2

5

x

80 3 10x

80 10

3 3x

Thus, the solution set for the given inequality is 80 10

,3 3

.

Question 6:

Solve the inequality 3 11

7 112

x

Solution 6:

3 117 11

2

x

14 3 11 22x

14 11 3 22 11x

3 3 11x

111

3x

Thus, the solution set for the given inequality is 11

1,3

.

Question 7:

Solve the inequalities and represent the solution graphically on number line: 5 1 24, 5 1 24x x

Solution 7:

5 1 24 5 25x x

5 ..... 1x

5 1 24 5 25x x

5 ...... 2x

From (1) and (2), it can be concluded that the solution set for the given system of inequalities is

5,5 . The solution of the given system of inequalities can be represented on number line as


Linear Inequalities


Question 8:

Solve the inequalities and represent the solution graphically on number line:

2 1 5x x , 3 2 2x x

Solution 8:

2 1 5 2 2 5 2 5 2x x x x x x

7 ....... 1x

3 2 2 3 6 2 3 2 6x x x x x x

4 4x

1 ....... 2x



Question 9:

Solve the following inequalities and represent the solution graphically on number line:

3 7 2 6 , 6 11 2x x x x

Solution 9:

3 7 2 6 3 7 2 12 3 2 12 7x x x x x x

5x ……… (1)

6 11 2 2 11 6x x x x

5 ...... 2x


5, . The solution of the given system of inequalities can be represented on number line as

Question 10:

Solve the inequalities and represent the solution graphically on number line:

5 2 7 3 2 3 0, 2 19 6 47x x x x

Solution 10:

5 2 7 3 2 3 0 10 35 6 9 0 4 44 0 4 44x x x x x x

11 ....... 1x


Linear Inequalities


2 19 6 47 19 47 6 2 28 4x x x x x

7 ...... 2x



Question 11:

A solution is to be kept between 68 F and 77 F . What is the range in temperature in degree

Celsius (C) if the Celsius/Fahrenheit (F) conversion formula is given by

932

5F C ?

Solution 11:

Since the solution is to be kept between 68 F and 77 ,68 77F F

Putting 9

32,5

F C we obtain

968 32 77

5C

968 32 77 32

5C

936 45

5C

5 536 45

9 9C

20 25C

Thus, the required range of temperature in degree Celsius is between 20 C and 25 C .

Question 12:

A solution of 8% boric acid is to be diluted by adding a 2% boric acid solution to it. The resulting

mixture is to be more than 4% but less than 6% boric acid. If we have 640 litres of the 8%

solution, how many litres of the 2% solution will have to be added?

Solution 12:

Let x litres of 2% boric acid solution is required to be added.

Then, total mixture 640 litresx

This resulting mixture is to be more than 4% but less than 6% boric acid.

2% 8% of 640 4% of 640 and 8% of 640 6% of 640x x x x

2% 8% of 640 4% of 640x x

2 8 4

640 640100 100 100

x x

2 5120 4 2560x x


Linear Inequalities


5120 2560 4 2x x

5120 2560 2x

2560 2x

1280 x

2% 8% of 640 6% of 640x x

2 8 6

640 640100 100 100

x x

2 5120 6 3840x x

5120 3840 6 2x x

5120 3840 6 2x x

1280 4x

320 x 320 1280x

Thus, the number of litres of 2% of boric acid solution that is to be added will have to be more

than 320 litres but less than 1280 litres.

Question 13:

How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that

the resulting mixture will contain more than 25% but less than 30% acid content?

Solution 13:

Let x litres of water is required to be added.

Then, total mixture 1125x litres

It is evident that the amount of acid contained in the resulting mixture is 45% of 1125 litres.

This resulting mixture will contain more than 25% but less than 30% acid content.

30% of 1125 45% of 1125x

And, 25% of 1125 45% of 1125x

30% of 1125 45% of 1125x

30 45

1125 1125100 100

x

30 1125 45 1125x

30 1125 30 45 1125x

30 45 1125 30 1125

30 45 30 1125x

15 1125562.5

30x

25% of 1125 45% of 1125x

25 45

1125 1125100 100

x

25 1125 45 1125x

25 1125 25 45 1125x


Linear Inequalities


25 45 1125 25 1125x

25 45 25 1125x

20 1125900

25x

562.5 900x

Thus, the required number of litres of water that is to be added will have to be more than 562.5

but less than 900.

Question 14:

IQ of a person is given by the formula 100MA

IQCA

,

Where MA is mental age and CA is chronological age. If 80 140IQ for a group of 12 years

old children, find the range of their mental age.

Solution 14:

It is given that for a group of 12 years old children,

80 140 .......IQ i

For a group of 12 years old children, CA 12 years

MAIQ 100

12

Putting this value of IQ in (i), we obtain

MA80 100 140

12

12 1280 MA 140

100 100

9.6 MA 16.8

Thus, the range of mental age of the group of 12 years old children is 9.6 MA 16.8 .


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